Looks like your off to a good start.
As for the foam core. I peel off the "paper" or whatever it is. One or both sides, depending on how I'm gonna use it. It does give the foam some strength. Once you peel off the paper you can use a pen to carve it. It's very easy to work with. I get the really cheap stuff from the dollar store. Very easy to peel off the paper.
You can also use warm water to soften the glue on the paper.
Here's a design and stonework carved into a floor of a piece I made.
